Equivalence of (quasi-)norms on a vector-valued function space and its applications to multilinear operators

In this paper we present (quasi-)norm equivalence on a vector-valued function space $L^p_A(l^q)$ and extend the to $p=\infty$ $0<q<\infty$ in scale of Triebel-Lizorkin space, motivated by Fraizer-Jawerth. By applying results, improve multilinear Hormander's multiplier theorem Tomita, that Grafakos-Si, boundedness results for bilinear pseudo-differential operators, given Koezuka-Tomita.